According to

> Fleischmann, Peter; Janiszczak, Ingo.
> [On the computation of conjugacy classes of Chevalley groups][1].
> *Appl. Alg. in Eng., Comm. and Comp.* 1996, 7(3), 221--234

the class number of ${\rm F}_4(q)$ is

  - $q^4 + 2q^3 + 6q^2 + 10q + 19$ if $q = 2^n$,

  - $q^4 + 2q^3 + 7q^2 + 15q + 30$ if $q = 3^n$, and

  - $q^4 + 2q^3 + 7q^2 + 15q + 31$ if $q = p^n$ where $p > 3$

(see page 233).

According to [Frank Lübeck's database][2] on finite groups of Lie type,
the class number of $^2{\rm F}_4(q^2)$ is $q^4+4q^2+17$.
In that database you also find class numbers for many other types of finite
groups of Lie type.

  [1]: http://link.springer.com/article/10.1007%2FBF01190331
  [2]: http://www.math.rwth-aachen.de/~Frank.Luebeck/chev/index.html?LANG=de